Purely magnetic spacetimes

TL;DR

The paper reviews the history and recent progress on the conjecture that purely magnetic vacuum spacetimes do not exist, and discusses approaches to prove this conjecture using the Newman-Penrose formalism.

Contribution

It summarizes previous work, presents new special case results, and proposes a method for a general proof of the non-existence conjecture.

Findings

Purely magnetic vacuum spacetimes are conjectured not to exist.

Recent advances support the conjecture in certain cases.

A new approach using Newman-Penrose formalism is proposed.

Abstract

Spacetimes in which the electric part of the Weyl tensor vanishes (relative to some timelike unit vector field) are said to be purely magnetic. Examples of purely magnetic spacetimes are known and are relatively easy to construct, if no restrictions are placed on the energy-momentum tensor. However it has long been conjectured that purely magnetic vacuum spacetimes (with or without a cosmological constant) do not exist. The history of this conjecture is reviewed and some advances made in the last year are described briefly. A generalisation of this conjecture first suggested for type D vacuum spacetimes by Ferrando and Saez is stated and proved in a number of special cases. Finally an approach to a general proof of the conjecture is described using the Newman-Penrose formalism based on a canonical null tetrad of the Weyl tensor.